130edo: Difference between revisions

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=== Subsets and supersets ===
=== Subsets and supersets ===
Since 130 factors into 2 × 5 × 13, 130edo has subset edos {{EDOs| 2, 5, 10, 13, 26, and 65 }}.
Since 130 factors into {{nowrap|2 × 5 × 13}}, 130edo has subset edos {{EDOs| 2, 5, 10, 13, 26, and 65 }}.


[[260edo]], which divides the edostep in two, provides a strong correction for the 29th harmonic.
[[260edo]], which divides the edostep in two, provides a strong correction for the 29th harmonic.
Line 324: Line 324:


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 3136/3125, 19683/19600
| 2401/2400, 3136/3125, 19683/19600
| {{mapping| 130 206 302 365 }}
| {{mapping| 130 206 302 365 }}
| -0.119
| &minus;0.119
| 0.311
| 0.311
| 3.37
| 3.37
Line 344: Line 336:
| 243/242, 441/440, 3136/3125, 4000/3993
| 243/242, 441/440, 3136/3125, 4000/3993
| {{mapping| 130 206 302 365 450 }}
| {{mapping| 130 206 302 365 450 }}
| -0.241
| &minus;0.241
| 0.370
| 0.370
| 4.02
| 4.02
Line 351: Line 343:
| 243/242, 351/350, 364/363, 441/440, 3136/3125
| 243/242, 351/350, 364/363, 441/440, 3136/3125
| {{mapping| 130 206 302 365 450 481 }}
| {{mapping| 130 206 302 365 450 481 }}
| -0.177
| &minus;0.177
| 0.367
| 0.367
| 3.98
| 3.98
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Note: temperaments supported by [[65edo|65et]] are not included.  
Note: temperaments supported by [[65edo|65et]] are not included.  


{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 434: Line 420:
|-
|-
| 2
| 2
| 54\130<br>(11\130)
| 54\130<br />(11\130)
| 498.46<br>(101.54)
| 498.46<br />(101.54)
| 4/3<br>(35/33)
| 4/3<br />(35/33)
| [[Bischismic]]
| [[Bischismic]]
|-
|-
| 5
| 5
| 27\130<br>(1\130)
| 27\130<br />(1\130)
| 249.23<br>(9.23)
| 249.23<br />(9.23)
| 81/70<br>(176/175)
| 81/70<br />(176/175)
| [[Hemipental]]
| [[Hemipental]]
|-
|-
| 10
| 10
| 27\130<br>(1\130)
| 27\130<br />(1\130)
| 249.23<br>(9.23)
| 249.23<br />(9.23)
| 15/13<br>(176/175)
| 15/13<br />(176/175)
| [[Decoid]]
| [[Decoid]]
|-
|-
| 10
| 10
| 54\130<br>(2\130)
| 54\130<br />(2\130)
| 498.46<br>(18.46)
| 498.46<br />(18.46)
| 4/3<br>(81/80)
| 4/3<br />(81/80)
| [[Decal]]
| [[Decal]]
|-
|-
| 26
| 26
| 54\130<br>(1\130)
| 54\130<br />(1\130)
| 498.46<br>(9.23)
| 498.46<br />(9.23)
| 4/3<br>(225/224)
| 4/3<br />(225/224)
| [[Bosonic]]
| [[Bosonic]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


== Scales ==
== Scales ==
{| class="wikitable"
{| class="wikitable"
|+14-tone temperament of "Narrative Wars"<br>as an example of using 130edo:
|+ style="font-size: 105%;" | 14-tone temperament of "Narrative Wars"<br />as an example of using 130edo:
|-
! Step
! Step
! Cents
! Cents
! Distance to the nearest JI interval<br>(selected ratios)
! Distance to the nearest JI interval<br />(selected ratios)
|-
|-
| 13 (13/130)
| 13 (13/130)
Line 490: Line 477:
| 9 (47/130)
| 9 (47/130)
| 433.846
| 433.846
| [[9/7]] (-1.238 ¢)
| [[9/7]] (&minus;1.238 ¢)
|-
|-
| 7 (54/130)
| 7 (54/130)
Line 502: Line 489:
| 9 (76/130)
| 9 (76/130)
| 701.538
| 701.538
| [[3/2]] (-0.417 ¢)
| [[3/2]] (&minus;0.417 ¢)
|-
|-
| 7 (83/130)
| 7 (83/130)
Line 514: Line 501:
| 5 (101/130)
| 5 (101/130)
| 932.308
| 932.308
| [[12/7]] (-0.821 ¢)
| [[12/7]] (&minus;0.821 ¢)
|-
|-
| 13 (114/130)
| 13 (114/130)
Line 522: Line 509:
| 7 (121/130)
| 7 (121/130)
| 1116.923
| 1116.923
| [[21/11]] (-2.540 ¢)
| [[21/11]] (&minus;2.540 ¢)
|-
|-
| 9 (130/130)
| 9 (130/130)
| 1200.000
| 1200.000
| [[Octave]] (2/1, ±0 ¢)
| [[Octave]] (2/1, 0 ¢)
|}
|}


Line 544: Line 531:
[[Category:Hemischis]]
[[Category:Hemischis]]
[[Category:Hemiwürschmidt]]
[[Category:Hemiwürschmidt]]
[[Category:Listen]]
[[Category:Sesquiquartififths]]
[[Category:Sesquiquartififths]]
[[Category:Listen]]

Revision as of 06:23, 16 November 2024

← 129edo 130edo 131edo →
Prime factorization 2 × 5 × 13
Step size 9.23077 ¢ 
Fifth 76\130 (701.538 ¢) (→ 38\65)
Semitones (A1:m2) 12:10 (110.8 ¢ : 92.31 ¢)
Consistency limit 15
Distinct consistency limit 15

Template:EDO intro

Theory

130edo is a zeta peak edo, a zeta peak integer edo, and a zeta integral edo but not a gap edo. It is distinctly consistent to the 15-odd-limit and is the first nontrivial edo to be consistent in the 14-odd-prime-sum-limit. As an equal temperament, it tempers out 2401/2400, 3136/3125, 6144/6125, and 19683/19600 in the 7-limit; 243/242, 441/440, 540/539, and 4000/3993 in the 11-limit; and 351/350, 364/363, 676/675, 729/728, 1001/1000, 1575/1573, 1716/1715, 2080/2079, 4096/4095, and 4225/4224 in the 13-limit. It can be used to tune a variety of temperaments, including hemiwürschmidt, sesquiquartififths, harry and hemischis. It also can be used to tune the rank-3 temperament jove, tempering out 243/242 and 441/440, plus 364/363 for the 13-limit and 595/594 for the 17-limit. It gives the optimal patent val for 11-limit hemiwürschmidt and sesquart and 13-limit harry.

Prime harmonics

Approximation of prime harmonics in 130edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.42 +1.38 +0.40 +2.53 -0.53 -3.42 -2.13 -0.58 +4.27 -0.42
Relative (%) +0.0 -4.5 +14.9 +4.4 +27.4 -5.7 -37.0 -23.1 -6.3 +46.2 -4.6
Steps
(reduced) 		130
(0) 		206
(76) 		302
(42) 		365
(105) 		450
(60) 		481
(91) 		531
(11) 		552
(32) 		588
(68) 		632
(112) 		644
(124)

Subsets and supersets

Since 130 factors into 2 × 5 × 13, 130edo has subset edos 2, 5, 10, 13, 26, and 65.

260edo, which divides the edostep in two, provides a strong correction for the 29th harmonic.

Intervals

Degree Cents Approximate Ratios
0 0.000 1/1
1 9.231 126/125, 144/143, 169/168, 176/175, 196/195, 225/224
2 18.462 78/77, 81/80, 91/90, 99/98, 100/99, 105/104, 121/120
3 27.692 56/55, 64/63, 65/64, 66/65
4 36.923 45/44, 49/48, 50/49, 55/54
5 46.154 36/35, 40/39
6 55.385 33/32
7 64.615 27/26, 28/27
8 73.846 25/24, 26/25
9 83.077 21/20, 22/21
10 92.308 135/128
11 101.538 35/33
12 110.769 16/15
13 120.000 15/14
14 129.231 14/13
15 138.462 13/12
16 147.692 12/11
17 156.923 35/32
18 166.154 11/10
19 175.385 72/65
20 184.615 10/9
21 193.846 28/25
22 203.077 9/8
23 212.308 44/39
24 221.538 25/22
25 230.769 8/7
26 240.000 55/48
27 249.231 15/13
28 258.462 64/55
29 267.692 7/6
30 276.923 75/64
31 286.154 13/11
32 295.385 32/27
33 304.615 25/21
34 313.846 6/5
35 323.077 65/54
36 332.308 40/33
37 341.538 39/32
38 350.769 11/9, 27/22
39 360.000 16/13
40 369.231 26/21
41 378.462 56/45
42 387.692 5/4
43 396.923 44/35
44 406.154 81/64
45 415.385 14/11
46 424.615 32/25
47 433.846 9/7
48 443.077 84/65, 128/99
49 452.308 13/10
50 461.538 64/49, 72/55
51 470.769 21/16
52 480.000 33/25
53 489.231 65/49
54 498.462 4/3
55 507.692 75/56
56 516.923 27/20
57 526.154 65/48
58 535.385 15/11
59 544.615 48/35
60 553.846 11/8
61 563.077 18/13
62 572.308 25/18
63 581.538 7/5
64 590.769 45/32
65 600.000 99/70, 140/99
… … …

Notation

Sagittal

Steps 0 1 2 3 4 5 6 7 8 9 10 11 12
Symbol

Regular temperament properties

Template:Comma basis begin |- | 2.3.5.7 | 2401/2400, 3136/3125, 19683/19600 | [⟨130 206 302 365]] | −0.119 | 0.311 | 3.37 |- | 2.3.5.7.11 | 243/242, 441/440, 3136/3125, 4000/3993 | [⟨130 206 302 365 450]] | −0.241 | 0.370 | 4.02 |- | 2.3.5.7.11.13 | 243/242, 351/350, 364/363, 441/440, 3136/3125 | [⟨130 206 302 365 450 481]] | −0.177 | 0.367 | 3.98 Template:Comma basis end

Rank-2 temperaments

Note: temperaments supported by 65et are not included.

Template:Rank-2 begin |- | 1 | 3\130 | 27.69 | 64/63 | Arch |- | 1 | 7\130 | 64.62 | 26/25 | Rectified hebrew |- | 1 | 9\130 | 83.08 | 21/20 | Sextilififths |- | 1 | 19\130 | 175.38 | 72/65 | Sesquiquartififths / sesquart |- | 1 | 21\130 | 193.85 | 28/25 | Hemiwürschmidt |- | 1 | 27\130 | 249.23 | 15/13 | Hemischis |- | 1 | 41\130 | 378.46 | 56/45 | Subpental |- | 2 | 6\130 | 55.38 | 33/32 | Septisuperfourth |- | 2 | 9\130 | 83.08 | 21/20 | Harry |- | 2 | 17\130 | 156.92 | 35/32 | Bison |- | 2 | 19\130 | 175.38 | 448/405 | Bisesqui |- | 2 | 54\130
(11\130) | 498.46
(101.54) | 4/3
(35/33) | Bischismic |- | 5 | 27\130
(1\130) | 249.23
(9.23) | 81/70
(176/175) | Hemipental |- | 10 | 27\130
(1\130) | 249.23
(9.23) | 15/13
(176/175) | Decoid |- | 10 | 54\130
(2\130) | 498.46
(18.46) | 4/3
(81/80) | Decal |- | 26 | 54\130
(1\130) | 498.46
(9.23) | 4/3
(225/224) | Bosonic Template:Rank-2 end Template:Orf

Scales

14-tone temperament of "Narrative Wars"
as an example of using 130edo:
Step Cents Distance to the nearest JI interval
(selected ratios)
13 (13/130) 120.000 15/14 (+0.557 ¢)
7 (20/130) 184.615 10/9 (+2.211 ¢)
9 (29/130) 267.692 7/6 (+0,821 ¢)
9 (38/130) 350.769 11/9 (+3.361 ¢)
9 (47/130) 433.846 9/7 (−1.238 ¢)
7 (54/130) 498.462 4/3 (+0.417 ¢)
13 (67/130) 618.462 10/7 (+0.974 ¢)
9 (76/130) 701.538 3/2 (−0.417 ¢)
7 (83/130) 766.154 14/9 (+1.238 ¢)
13 (96/130) 886.154 5/3 (+1.795 ¢)
5 (101/130) 932.308 12/7 (−0.821 ¢)
13 (114/130) 1052.308 11/6 (+2.945 ¢)
7 (121/130) 1116.923 21/11 (−2.540 ¢)
9 (130/130) 1200.000 Octave (2/1, 0 ¢)

Music

See also: Category:130edo tracks
birdshite stalactite
Sevish
Gene Ward Smith
Retrieved from "https://en.xen.wiki/index.php?title=130edo&oldid=165993"